Static fluctuations of a thick one-dimensional interface in the 1+1 directed polymer formulation

Experimental realizations of a one-dimensional (1D) interface always exhibit a finite microscopic width xi > 0; its influence is erased by thermal fluctuations at sufficiently high temperatures, but turns out to be a crucial ingredient for the description of the interface fluctuations below a characteristic temperature T-c (xi). Exploiting the exact mapping between the static 1D interface and a 1 + 1 directed polymer (DP) growing in a continuous space, we study analytically both the free-energy and geometrical fluctuations of a DP, at finite temperature T, with a short-range elasticity and submitted to a quenched random-bond Gaussian disorder of finite correlation length xi. We derive the exact time-evolution equations of the disorder free energy (F) over bar (t, y), which encodes the microscopic disorder integrated by the DP up to a growing time t and an endpoint position y, its derivative eta(t, y), and their respective two-point correlators (C) over bar (t, y) and (R) over bar (t, y). We compute the exact solution of its linearized evolution (R) over bar (lin)(t, y) and we combine its qualitative behavior and the asymptotic properties known for an uncorrelated disorder (xi = 0) to justify the construction of a "toy model" leading to a simple description of the DP properties. This model is characterized by Gaussian Brownian-type free-energy fluctuations, correlated at small vertical bar y vertical bar less than or similar to xi, and of amplitude (D) over tilde (infinity)(T,xi). We present an extended scaling analysis of the roughness, supported by saddle-point arguments on its path-integral representation, which predicts (D) over tilde (infinity) similar to 1/T at high temperatures and (D) over tilde (infinity) 1/T-c (xi) at low temperatures. We identify the connection between the temperature-induced crossover of (D) over tilde (infinity)(T,xi) and the full replica symmetry breaking in previous Gaussian variational method (GVM) computations. In order to refine our toy model with respect to finite-time geometrical fluctuations, we propose an effective time-dependent amplitude (D) over tilde (t). Finally, we discuss the consequences of the low-temperature regime for two experimental realizations of Kardar-Parisi-Zhang interfaces, namely, the static and quasistatic behavior of magnetic domain walls and the high-velocity steady-state dynamics of interfaces in liquid crystals. DOI: 10.1103/PhysRevE.87.042406